When L/K is separable, the trace provides a duality theory via the trace form: the map from L × L to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory in the theory of the different ideal.
The trace form for a finite degree field extension L/K has non-negative signature for any field ordering of K. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for algebraic number fields K.
Read more about this topic: Field Trace
Famous quotes containing the words trace and/or form:
“Emancipation should make it possible for woman to be human in the truest sense. Everything within her that craves assertion and activity should reach its fullest expression; all artificial barriers should be broken, and the road towards greater freedom cleared of every trace of centuries of submission and slavery.”
—Emma Goldman (1869–1940)
“After all, what is reading but a vice, like drink or venery or any other form of excessive self-indulgence? One reads to tickle and amuse one’s mind; one reads, above all, to prevent oneself thinking.”
—Aldous Huxley (1894–1963)